Combining link-tracing sampling and cluster sampling to estimate the size of a hidden population in presence of heterogeneous link-probabilities 6. Monte Carlo studies

6.1 Populations constructed using artificial data

We constructed four artificial populations; a description of each one is presented in Table 6.1. Notice that in Populations I, III and IV the N=150 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaey ypa0JaaGymaiaaiwdacaaIWaaaaa@3C72@ values of the m i s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohaaaa@3C34@ were obtained by sampling from a Poisson distribution, whereas in Population II by sampling from a zero truncated negative binomial distribution. In addition, in Populations I and II, the link-probabilities p i j ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaaaa@3DDD@ were generated by the Rasch model (3.2). In Population III they were generated by that model but the random effects β j ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa aaa@3D9B@ were obtained by sampling from a scaled Student’s T distribution with six degrees of freedom and unit-variance instead of by sampling from the standard normal distribution. Finally, in Population IV, the p i j ( k ) s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaGqaaOGaa8xgGiaabohaaaa@3FA0@ were generated by the following latent class model proposed by Pledger (2000) in the context of capture-recapture studies: p ij ( k ) =exp[ μ ( k ) + α i ( k ) + β j ( k ) + ( αβ ) ij ( k ) ]/ { 1+exp[ μ ( k ) + α i ( k ) + β j ( k ) + ( αβ ) ij ( k ) ] } , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaWcgaqaai aadchadaqhaaWcbaGaamyAaiaadQgaaeaadaqadaqaaiaadUgaaiaa wIcacaGLPaaaaaGccqGH9aqpciGGLbGaaiiEaiaacchadaWadaqaai abeY7aTnaaCaaaleqabaWaaeWaaeaacaWGRbaacaGLOaGaayzkaaaa aOGaey4kaSIaeqySde2aa0baaSqaaiaadMgaaeaadaqadaqaaiaadU gaaiaawIcacaGLPaaaaaGccqGHRaWkcqaHYoGydaqhaaWcbaGaamOA aaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaakiabgUcaRmaabm aabaGaeqySdeMaeqOSdigacaGLOaGaayzkaaWaa0baaSqaaiaadMga caWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzkaaaaaaGccaGLBb GaayzxaaaabaWaaiWaaeaacaaIXaGaey4kaSIaciyzaiaacIhacaGG WbWaamWaaeaacqaH8oqBdaahaaWcbeqaamaabmaabaGaam4AaaGaay jkaiaawMcaaaaakiabgUcaRiabeg7aHnaaDaaaleaacaWGPbaabaWa aeWaaeaacaWGRbaacaGLOaGaayzkaaaaaOGaey4kaSIaeqOSdi2aa0 baaSqaaiaadQgaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaGc cqGHRaWkdaqadaqaaiabeg7aHjabek7aIbGaayjkaiaawMcaamaaDa aaleaacaWGPbGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMca aaaaaOGaay5waiaaw2faaaGaay5Eaiaaw2haaaaacaGGSaaaaa@82D9@ i=1,,n;j=1,2, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbGaey ypa0JaaGymaiaacYcacqWIMaYscaaISaGaamOBaiaacUdacaWGQbGa eyypa0JaaGymaiaacYcacaaIYaGaaiilaaaa@441A@ and k=1,2. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGRbGaey ypa0JaaGymaiaacYcacaaIYaGaaiOlaaaa@3D34@ In this model the people in U k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS baaSqaaiaadUgaaeqaaaaa@3A5B@ is divided into two latent classes ( j=1,2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai aadQgacqGH9aqpcaaIXaGaaiilaiaaikdaaiaawIcacaGLPaaaaaa@3E0A@ according to their propensities to be linked to the sampled clusters. The probability that a randomly person in U k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS baaSqaaiaadUgaaeqaaaaa@3A5B@ is in class j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbaaaa@3954@ is p j ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadQgaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaaa aa@3CEF@ and the p i j ( k ) s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaGqaaOGaa8xgGiaabohaaaa@3FA0@ are the same for all the people in the class j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbGaai Olaaaa@3A06@

The simulation experiment was carried out by repeatedly selecting r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbaaaa@395C@ samples from each population by using the sampling design described in Section 2 with initial sample size n=15. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbGaey ypa0JaaGymaiaaiwdacaGGUaaaaa@3C8A@ Thus, each time that the value m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS baaSqaaiaadMgaaeqaaaaa@3A71@ was included in an initial sample, the value x i j ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG4bWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaaaa@3DE5@ was obtained by sampling from the Bernoulli distribution with mean p i j ( k ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaOGaaiOlaaaa@3E99@ Because of the values assigned to n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbaaaa@3958@ and to the parameters that appear in the expression of p i j ( k ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaOGaaiilaaaa@3E97@ the resulting sampling rates were f 1 0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS baaSqaaiaaigdaaeqaaOGaeyisISRaaGimaiaai6cacaaI1aaaaa@3E23@ and f 2 0.4. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS baaSqaaiaaikdaaeqaaOGaeyisISRaaGimaiaai6cacaaI0aGaaiOl aaaa@3ED5@ It is worth noting that the characteristics of the populations and samples considered in this study were not motivated by the ones of an actual study since this sampling design has not been applied yet. Thus, the populations and samples were constructed only with the purpose of analyzing the performance of the proposed point and interval estimators.

Table 6.1
Parameters of the simulated populations
Table summary
This table displays the results of Parameters of the simulated populations. The information is grouped by Population I (appearing as row headers), Population II, Population III and Population IV (appearing as column headers).
Population I Population II Population III Population IV
N = 150 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaey ypa0JaaGymaiaaiwdacaaIWaaaaa@3F41@ N = 150 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaey ypa0JaaGymaiaaiwdacaaIWaaaaa@3F41@ N = 150 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaey ypa0JaaGymaiaaiwdacaaIWaaaaa@3F41@ N = 150 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaey ypa0JaaGymaiaaiwdacaaIWaaaaa@3F41@
M i Poisson MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadMgaaeqaaebbfv3ySLgzGueE0jxyaGqbaOGae8hpIOJa aeiuaiaab+gacaqGPbGaae4CaiaabohacaqGVbGaaeOBaaaa@4963@ M i Zero trunc . neg . binomial MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadMgaaeqaaebbfv3ySLgzGueE0jxyaGqbaOGae8hpIOJa aeOwaiaabwgacaqGYbGaae4BaiaabccacaqG0bGaaeOCaiaabwhaca qGUbGaae4yaiaab6cacaqGGaGaaeOBaiaabwgacaqGNbGaaeOlaiaa bccacaqGIbGaaeyAaiaab6gacaqGVbGaaeyBaiaabMgacaqGHbGaae iBaaaa@58BB@ M i Poisson MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadMgaaeqaaebbfv3ySLgzGueE0jxyaGqbaOGae8hpIOJa aeiuaiaab+gacaqGPbGaae4CaiaabohacaqGVbGaaeOBaaaa@4963@ M i Poisson MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadMgaaeqaaebbfv3ySLgzGueE0jxyaGqbaOGae8hpIOJa aeiuaiaab+gacaqGPbGaae4CaiaabohacaqGVbGaaeOBaaaa@4963@
E ( M i ) = 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGfbWaae WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa eyypa0JaaGioaaaa@4145@ E ( M i ) = 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGfbWaae WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa eyypa0JaaGioaaaa@4145@ E ( M i ) = 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGfbWaae WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa eyypa0JaaGioaaaa@4145@ E ( M i ) = 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGfbWaae WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa eyypa0JaaGioaaaa@4145@
V ( M i ) = 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaae WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa eyypa0JaaGioaaaa@4156@ V ( M i ) = 24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaae WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa eyypa0JaaGOmaiaaisdaaaa@420E@ V ( M i ) = 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaae WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa eyypa0JaaGioaaaa@4156@ V ( M i ) = 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaae WaaeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGa eyypa0JaaGioaaaa@4156@
τ 1 = 1,209 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccqGH9aqpcaqGXaGaaeilaiaabkdacaqG WaGaaeyoaaaa@4277@ τ 1 = 1,208 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIXaGaaiilaiaaikdacaaI WaGaaGioaaaa@4293@ τ 1 = 1,209 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIXaGaaiilaiaaikdacaaI WaGaaGyoaaaa@4294@ τ 1 = 1,209 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIXaGaaiilaiaaikdacaaI WaGaaGyoaaaa@4294@
τ 2 = 400 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI0aGaaGimaiaaicdaaaa@4123@ τ 2 = 400 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI0aGaaGimaiaaicdaaaa@4123@ τ 2 = 400 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI0aGaaGimaiaaicdaaaa@4123@ τ 2 = 400 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI0aGaaGimaiaaicdaaaa@4123@
τ = 1,609 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDcq GH9aqpcaqGXaGaaeilaiaabAdacaqGWaGaaeyoaaaa@418A@ τ = 1,608 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDcq GH9aqpcaaIXaGaaiilaiaaiAdacaaIWaGaaGioaaaa@41A6@ τ = 1,609 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDcq GH9aqpcaaIXaGaaiilaiaaiAdacaaIWaGaaGyoaaaa@41A7@ τ = 1,609 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDcq GH9aqpcaaIXaGaaiilaiaaiAdacaaIWaGaaGyoaaaa@41A7@
α i ( k ) = c k M i 1 / 4 + 0.001 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda qhaaWcbaGaamyAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa kiabg2da9maalaaabaGaam4yamaaBaaaleaacaWGRbaabeaaaOqaai aad2eadaqhaaWcbaGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisda aaaaaOGaey4kaSIaaGimaiaai6cacaaIWaGaaGimaiaaigdaaaaaaa@4B9E@ α i ( k ) = c k M i 1 / 4 + 0.001 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda qhaaWcbaGaamyAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa kiabg2da9maalaaabaGaam4yamaaBaaaleaacaWGRbaabeaaaOqaai aad2eadaqhaaWcbaGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisda aaaaaOGaey4kaSIaaGimaiaai6cacaaIWaGaaGimaiaaigdaaaaaaa@4B9E@ α i ( k ) = c k M i 1 / 4 + 0.001 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda qhaaWcbaGaamyAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa kiabg2da9maalaaabaGaam4yamaaBaaaleaacaWGRbaabeaaaOqaai aad2eadaqhaaWcbaGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisda aaaaaOGaey4kaSIaaGimaiaai6cacaaIWaGaaGimaiaaigdaaaaaaa@4B9E@ α i ( k ) = 12 M i 1 / 2 + 0.001 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda qhaaWcbaGaamyAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa kiabg2da9maalaaabaGaeyOeI0IaaGymaiaaikdaaeaacaWGnbWaa0 baaSqaaiaadMgaaeaadaWcgaqaaiaaigdaaeaacaaIYaaaaaaakiab gUcaRiaaicdacaaIUaGaaGimaiaaicdacaaIXaaaaaaa@4BF2@
c 1 = 5.45 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS baaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0IaaGynaiaai6cacaaI 0aGaaGynaaaa@41F4@ c 1 = 5.45 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS baaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0IaaGynaiaai6cacaaI 0aGaaGynaaaa@41F4@ c 1 = 5.45 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS baaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0IaaGynaiaai6cacaaI 0aGaaGynaaaa@41F4@ μ (1) =1.1; μ (2) =1.2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda ahaaWcbeqaaiaacIcacaaIXaGaaiykaaaakiabg2da9iabgkHiTiaa igdacaaIUaGaaGymaiaacUdacaaMe8UaaGPaVlabeY7aTnaaCaaale qabaGaaiikaiaaikdacaGGPaaaaOGaeyypa0JaeyOeI0IaaGymaiaa i6cacaaIYaaaaa@4F51@
c 2 = 5.85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS baaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGynaiaai6cacaaI 4aGaaGynaaaa@41F9@ c 2 = 5.85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS baaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGynaiaai6cacaaI 4aGaaGynaaaa@41F9@ c 2 = 5.85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS baaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGynaiaai6cacaaI 4aGaaGynaaaa@41F9@ β 1 (k) =1.5; β 2 (k) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda qhaaWcbaGaaGymaaqaaiaacIcacaWGRbGaaiykaaaakiabg2da9iaa igdacaGGUaGaaGynaiaacUdacaaMe8UaaGPaVlabek7aInaaDaaale aacaaIYaaabaGaaiikaiaadUgacaGGPaaaaOGaeyypa0JaaGimaaaa @4DB6@
β j ( k ) N ( 0,1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa rqqr1ngBPrgifHhDYfgaiuaakiab=XJi6iaad6eadaqadaqaaiaaic dacaaISaGaaGymaaGaayjkaiaawMcaaaaa@4AB4@ β j ( k ) N ( 0 , 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa rqqr1ngBPrgifHhDYfgaiuaakiab=XJi6iaad6eadaqadaqaaiaaic dacaGGSaGaaGymaaGaayjkaiaawMcaaaaa@4AAE@ β j ( k ) 2 / 3 T 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa rqqr1ngBPrgifHhDYfgaiuaakiab=XJi6maakaaabaWaaSGbaeaaca aIYaaabaGaaG4maaaaaSqabaGccaaMc8UaamivamaaBaaaleaacaaI 2aaabeaaaaa@4B31@ ( α β ) i j ( k ) N ( 0 , 1.25 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai abeg7aHjabek7aIbGaayjkaiaawMcaamaaDaaaleaacaWGPbGaamOA aaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaarqqr1ngBPrgifH hDYfgaiuaakiab=XJi6iaad6eadaqadaqaaiaaicdacaGGSaGaaGym aiaac6cacaaIYaGaaGynamaaCaaaleqabaGaaGOmaaaaaOGaayjkai aawMcaaaaa@51E4@
p 1 ( k ) = 0.3 = 1 p 2 ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrVfFnuz0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaaigdaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaGc cqGH9aqpcaaIWaGaaiOlaiaaiodacqGH9aqpcaaIXaGaeyOeI0Iaam iCamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaaaa@49C8@

 

From each sample the following estimators of τ 1 , τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccaGGSaGaeqiXdq3aaSbaaSqaaiaaikda aeqaaaaa@3E78@ and τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa a@3A2A@ were considered: the proposed UMLEs τ ^ 1 ( U ) , τ ^ 2 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaOGaaiilaiqbes8a0zaajaWaa0baaSqaaiaaikdaaeaadaqada qaaiaadwfaaiaawIcacaGLPaaaaaaaaa@4360@ and τ ^ ( U ) ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaCaaaleqabaWaaeWaaeaacaWGvbaacaGLOaGaayzkaaaaaOGa ai4oaaaa@3D93@ the proposed CMLEs τ ^ 1 ( C ) , τ ^ 2 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaOGaaiilaiqbes8a0zaajaWaa0baaSqaaiaaikdaaeaadaqada qaaiaadoeaaiaawIcacaGLPaaaaaaaaa@433C@ and τ ^ ( C ) ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaCaaaleqabaWaaeWaaeaacaWGdbaacaGLOaGaayzkaaaaaOGa ai4oaaaa@3D81@ the MLEs τ ˜ 1 , τ ˜ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga acamaaBaaaleaacaaIXaaabeaakiaacYcacuaHepaDgaacamaaBaaa leaacaaIYaaabeaaaaa@3E96@ and τ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga acaaaa@3A39@ proposed by Félix-Medina and Thompson (2004) and derived under the assumption of homogeneous link-probabilities, and the Bayesian-assisted estimators τ 1 , τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga afamaaBaaaleaacaaIXaaabeaakiaacYcacuaHepaDgaafamaaBaaa leaacaaIYaaabeaaaaa@3EAE@ and τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga afaaaa@3A45@ proposed by Félix-Medina and Monjardin (2006), derived also under the homogeneity assumption and using the following initial distributions for τ 1 , τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccaGGSaGaeqiXdq3aaSbaaSqaaiaaikda aeqaaaaa@3E78@ and α i ( k ) =ln[ p i ( k ) / ( 1 p i ( k ) ) ], MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda qhaaWcbaGaamyAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa kiabg2da9iGacYgacaGGUbWaamWaaeaadaWcgaqaaiaadchadaqhaa WcbaGaamyAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaaaOqa amaabmaabaGaaGymaiabgkHiTiaadchadaqhaaWcbaGaamyAaaqaam aabmaabaGaam4AaaGaayjkaiaawMcaaaaaaOGaayjkaiaawMcaaaaa aiaawUfacaGLDbaacaGGSaaaaa@4F9B@ where p i ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgaaeaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaaaa aa@3CEE@ is given by (3.2), but setting β j ( k ) =0: ξ( τ 1 | λ 1 ) ( N λ 1 ) τ 1 / τ 1 ! MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa kiabg2da9iaaicdacaGG6aWaaSGbaeaacqaH+oaEdaqadaqaamaaei aabaGaeqiXdq3aaSbaaSqaaiaaigdaaeqaaaGccaGLiWoacqaH7oaB daWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaacqGHDisTdaqada qaaiaad6eacqaH7oaBdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGL PaaadaahaaWcbeqaaiabes8a0naaBaaameaacaaIXaaabeaaaaaake aacqaHepaDdaWgaaWcbaGaaGymaaqabaGccaGGHaaaaaaa@5736@ and ξ ( λ 1 ) λ 1 a 1 1 exp ( b 1 λ 1 ) ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH+oaEda qadaqaaiabeU7aSnaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMca aiabg2Hi1kabeU7aSnaaDaaaleaacaaIXaaabaGaamyyamaaBaaame aacaaIXaaabeaaliabgkHiTiaaigdaaaGcciGGLbGaaiiEaiaaccha daqadaqaaiabgkHiTiaadkgadaWgaaWcbaGaaGymaaqabaGccqaH7o aBdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaacaGG7aaaaa@508A@ ξ ( τ 2 | λ 2 ) λ 2 τ 2 / τ 2 ! MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaWcgaqaai abe67a4naabmaabaGaeqiXdq3aaSbaaSqaaiaaikdaaeqaaOWaaqqa aeaacqaH7oaBdaWgaaWcbaGaaGOmaaqabaaakiaawEa7aaGaayjkai aawMcaaiabg2Hi1kabeU7aSnaaDaaaleaacaaIYaaabaGaeqiXdq3a aSbaaWqaaiaaikdaaeqaaaaaaOqaaiabes8a0naaBaaaleaacaaIYa aabeaakiaacgcaaaaaaa@4CE9@ and ξ ( λ 2 ) λ 2 a 2 1 exp ( b 2 λ 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH+oaEda qadaqaaiabeU7aSnaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMca aiabg2Hi1kabeU7aSnaaDaaaleaacaaIYaaabaGaamyyamaaBaaame aacaaIYaaabeaaliabgkHiTiaaigdaaaGcciGGLbGaaiiEaiaaccha daqadaqaaiabgkHiTiaadkgadaWgaaWcbaGaaGOmaaqabaGccqaH7o aBdaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaacaGGSaaaaa@5080@ and ξ ( α i ( k ) | θ k ) exp [ ( α i ( k ) θ k ) 2 / 2 σ k 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH+oaEda qadaqaamaaeiaabaGaeqySde2aa0baaSqaaiaadMgaaeaadaqadaqa aiaadUgaaiaawIcacaGLPaaaaaaakiaawIa7aiabeI7aXnaaBaaale aacaWGRbaabeaaaOGaayjkaiaawMcaaiabg2Hi1kGacwgacaGG4bGa aiiCamaadmaabaWaaSGbaeaacqGHsisldaqadaqaaiabeg7aHnaaDa aaleaacaWGPbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzkaaaaaOGa eyOeI0IaeqiUde3aaSbaaSqaaiaadUgaaeqaaaGccaGLOaGaayzkaa WaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaiabeo8aZnaaDaaaleaa caWGRbaabaGaaGOmaaaaaaaakiaawUfacaGLDbaaaaa@5C94@ and ξ ( θ k ) exp [ ( θ k μ k ) 2 / 2 γ k 2 ] , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH+oaEda qadaqaaiabeI7aXnaaBaaaleaacaWGRbaabeaaaOGaayjkaiaawMca aiabg2Hi1kGacwgacaGG4bGaaiiCamaadmaabaWaaSGbaeaacqGHsi sldaqadaqaaiabeI7aXnaaBaaaleaacaWGRbaabeaakiabgkHiTiab eY7aTnaaBaaaleaacaWGRbaabeaaaOGaayjkaiaawMcaamaaCaaale qabaGaaGOmaaaaaOqaaiaaikdacqaHZoWzdaqhaaWcbaGaam4Aaaqa aiaaikdaaaaaaaGccaGLBbGaayzxaaGaaiilaaaa@53F4@ where a 1 =1.0, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGHbWaaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGymaiaac6cacaaIWaGaaiil aaaa@3E19@ b 1 =0.1, a 2 =6.0, b 2 =0.01, μ k =3.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGIbWaaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaiaai6cacaaIXaGaaiil aiaadggadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI2aGaaGOlai aaicdacaGGSaGaamOyamaaBaaaleaacaaIYaaabeaakiabg2da9iaa icdacaaIUaGaaGimaiaaigdacaGGSaGaeqiVd02aaSbaaSqaaiaadU gaaeqaaOGaeyypa0JaeyOeI0IaaG4maiaai6cacaaI1aaaaa@5159@ and σ k 2 = γ k 2 =9.0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaam4AaaqaaiaaikdaaaGccqGH9aqpcqaHZoWzdaqhaaWc baGaam4AaaqaaiaaikdaaaGccqGH9aqpcaaI5aGaaiOlaiaaicdaca GGUaaaaa@4482@ These values assigned to the parameters of the initial distributions made them practically non-informative. The Gaussian quadrature approximations (3.3) and (3.4) to the probabilities π x ( k ) ( α k , σ k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda qhaaWcbaGaaCiEaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa kmaabmaabaGaaCySdmaaBaaaleaacaWGRbaabeaakiaaiYcacqaHdp WCdaWgaaWcbaGaam4AaaqabaaakiaawIcacaGLPaaaaaa@455E@ and π x ( A i ) ( α 1 i , σ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda qhaaWcbaGaaCiEaaqaamaabmaabaGaamyqamaaBaaameaacaWGPbaa beaaaSGaayjkaiaawMcaaaaakmaabmaabaGaaCySdmaaDaaaleaaca aIXaaabaGaeyOeI0IaamyAaaaakiaaiYcacqaHdpWCdaWgaaWcbaGa aGymaaqabaaakiaawIcacaGLPaaaaaa@47CC@ were computed using q=40 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGXbGaey ypa0JaaGinaiaaicdaaaa@3BD9@ terms.

The performance of an estimator τ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcaaaa@3A3A@ of τ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDca GGSaaaaa@3ADA@ say, was evaluated by means of its relative bias ( r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaiikaiaabk hacqGHsislaaa@3AE3@ bias), the square root of its relative mean square error ( r mse ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada GcaaqaaiaabkhacqGHsislcaqGTbGaae4CaiaabwgaaSqabaaakiaa wIcacaGLPaaacaGGSaaaaa@3F63@ and the medians of its relative estimation errors (mdre) and its absolute relative estimation errors (mdare) defined by rbias = 1 r ( τ ^ i τ )/ ( rτ ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk HiTiaabkgacaqGPbGaaeyyaiaabohadaWcgaqaaiabg2da9maaqada beWcbaGaaGymaaqaaiaadkhaa0GaeyyeIuoakmaabmaabaGafqiXdq NbaKaadaWgaaWcbaGaamyAaaqabaGccqGHsislcqaHepaDaiaawIca caGLPaaaaeaadaqadaqaaiaadkhacqaHepaDaiaawIcacaGLPaaaca GGSaaaaaaa@4EE6@ rmse = 1 r ( τ ^ i τ ) 2 / ( r τ 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGcaaqaai aabkhacqGHsislcaqGTbGaae4CaiaabwgaaSqabaGccqGH9aqpdaGc aaqaamaalyaabaWaaabmaeqaleaacaaIXaaabaGaamOCaaqdcqGHri s5aOWaaeWaaeaacuaHepaDgaqcamaaBaaaleaacaWGPbaabeaakiab gkHiTiabes8a0bGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaO qaamaabmaabaGaamOCaiabes8a0naaCaaaleqabaGaaGOmaaaaaOGa ayjkaiaawMcaaaaaaSqabaGccaGGSaaaaa@5049@ mdre = median{ ( τ ^ i τ )/τ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiaab2gacaqGKbGaaeOCaiaabwgacaqGGaGaaeypaiaabccacaqG TbGaaeyzaiaabsgacaqGPbGaaeyyaiaab6gapaWaaiWaaeaadaWcga qaamaabmaabaGafqiXdqNbaKaadaWgaaWcbaGaamyAaaqabaGccqGH sislcqaHepaDaiaawIcacaGLPaaaaeaacqaHepaDaaaacaGL7bGaay zFaaaaaa@4F0E@ and mdrae = median{ | ( τ ^ i τ )/τ | }, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa Wdbiaab2gacaqGKbGaaeOCaiaabggacaqGLbGaaeiiaiaab2dacaqG GaGaaeyBaiaabwgacaqGKbGaaeyAaiaabggacaqGUbWdamaacmaaba WaaqWaaeaadaWcgaqaamaabmaabaGafqiXdqNbaKaadaWgaaWcbaGa amyAaaqabaGccqGHsislcqaHepaDaiaawIcacaGLPaaaaeaacqaHep aDaaaacaGLhWUaayjcSdaacaGL7bGaayzFaaGaaiilaaaa@53C4@ where τ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaBaaaleaacaWGPbaabeaaaaa@3B54@ was the value of τ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcaaaa@3A3A@ obtained in the i th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbWaaW baaSqabeaacaqG0bGaaeiAaaaaaaa@3B62@ sample, which in the case of the point estimators was 10,000.

We also considered the following 95% CIs for the τ s : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDie aacaWFzaIaae4CaiaacQdaaaa@3CA1@ the proposed PLCIs and adjusted for extra-Poisson variation PLCIs; the proposed bootstrap CIs based on B=100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGcbGaey ypa0JaaGymaiaaicdacaaIWaaaaa@3C61@ bootstrap samples and constructed assuming a lognormal distribution for τ ^ ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcaiabgkHiTiabe27aUbaa@3CDF@ and estimating V ^ ( τ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGcaaqaai qadAfagaqcamaabmaabaGafqiXdqNbaKaaaiaawIcacaGLPaaaaSqa baaaaa@3CC9@ by Huber’s proposal 2 estimator of scale with tuning value d=1.5; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGKbGaey ypa0JaaGymaiaac6cacaaI1aGaai4oaaaa@3D3F@ the design-based Wald CIs obtained from the MLEs τ ˜ 1 , τ ˜ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga acamaaBaaaleaacaaIXaaabeaakiaacYcacuaHepaDgaacamaaBaaa leaacaaIYaaabeaaaaa@3E96@ and τ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga acaaaa@3A39@ and proposed by Félix-Medina and Thompson (2004), and the design-based Wald CIs obtained from the Bayesian estimators τ 1 , τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga afamaaBaaaleaacaaIXaaabeaakiaacYcacuaHepaDgaafamaaBaaa leaacaaIYaaabeaaaaa@3EAE@ and τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga afaaaa@3A45@ and proposed by Félix-Medina and Monjardin (2006). It is worth noting that the PLCIs and adjusted PLCI were computed using the Venzon and Moolgavkar’s (1988) method or an algorithm based on the definition of a PLCI when the first method failed to find the endpoints of the intervals.

The performance of a CI was evaluated by its coverage probability (cp), the mean of its relative lengths (mrl) and the median of its relative lengths (mdrl) defined as the proportion of samples in which the parameter was contained in the interval and the the mean and the median of the lengths of the intervals divided by the value of the parameter, respectively. Since carrying out a simulation study on the CIs using a large number of replicated samples is a very time consuming task, we evaluated the performance of the PLCIs for τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaaaaa@3B11@ and τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaaaaa@3B12@ using r=1,000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbGaey ypa0JaaGymaiaacYcacaaIWaGaaGimaiaaicdaaaa@3DFB@ samples; that of the PLCIs for τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa a@3A2A@ using r=500 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbGaey ypa0JaaGynaiaaicdacaaIWaaaaa@3C95@ samples and that of the bootstrap CIs using r=250 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbGaey ypa0JaaGOmaiaaiwdacaaIWaaaaa@3C97@ samples. The performance of the CIs based on the estimators derived under the homogeneity assumption was evaluated by using r=10,000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbGaey ypa0JaaGymaiaaicdacaGGSaGaaGimaiaaicdacaaIWaaaaa@3EB5@ samples. The numerical study was carried out using the R software programming language [R Development Core Team (2013)].

The results of the study on the estimators of the population sizes are shown in Table 6.2 and in Figures 6.1 and 6.2. The main outcomes are the following. The distributions of the estimators UMLE τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaaaa@3D85@ and CMLE τ ^ 1 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaaaa@3D73@ were more or less symmetrical about τ 1 ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccaGG7aaaaa@3BDA@ thus, the two measures of bias ( r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaiikaiaabk hacqGHsislaaa@3AE3@ bias and mdre) showed similar values, as well as the two measures of variability ( r mse MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaiikamaaka aabaGaaeOCaiabgkHiTiaab2gacaqGZbGaaeyzaaWcbeaaaaa@3DCC@ and mdare). Both of these estimators performed acceptably well, except in Population III where the estimator τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaaaa@3D85@ presented moderate problems of bias and τ ^ 1 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaaaa@3D73@ showed something more serious problems of bias. The distributions of the estimators τ ^ 2 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaaaa@3D86@ and τ ^ 2 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaaaa@3D74@ were skewed to the right with very long tails. This caused that the values of their r bias MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk HiTiaabkgacaqGPbGaaeyyaiaabohaaaa@3DE2@ and r mse MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk HiTiaab2gacaqGZbGaaeyzaaaa@3D05@ tended to be large. However, in terms of the medians of their relative errors (mdre), these estimators presented moderate problems of bias in Populations I and III and serious problems in Population IV. In terms of the medians of their absolute relative errors (mdare) these estimators showed moderate problems of instability in the first three populations and serious problems in the fourth population. The distributions of the estimators τ ^ ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaCaaaleqabaWaaeWaaeaacaWGvbaacaGLOaGaayzkaaaaaaaa @3CCA@ and τ ^ ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaCaaaleqabaWaaeWaaeaacaWGdbaacaGLOaGaayzkaaaaaaaa @3CB8@ were similar to those of the estimators τ ^ 2 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaaaa@3D86@ and τ ^ 2 ( C ) ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaOGaai4oaaaa@3E3D@ thus, the quantities r bias MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk HiTiaabkgacaqGPbGaaeyyaiaabohaaaa@3DE2@ and r mse MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca qGYbGaeyOeI0IaaeyBaiaabohacaqGLbaaleqaaaaa@3D20@ were more sensitive to large values than the quantities mdre and mdare. Both of these estimators performed acceptably well in Populations I, II and IV; although in this last population the values of their r mse MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca qGYbGaeyOeI0IaaeyBaiaabohacaqGLbaaleqaaaaa@3D20@ were large because of the reasons previously indicated. In Population III both estimators presented problems of bias.

Although the deviation from the assumed Poisson distribution of the M i s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohaaaa@3C14@ increased the variability of all the proposed estimators, the increments were not large so that we consider that they have some robust properties against deviations from this assumption. The proposed estimators of τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaaaaa@3B11@ and τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa a@3A2A@ were in addition robust to the deviation from the assumed Rasch model for the p i j ( 1 ) s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaaIXaaacaGLOaGaayzk aaaaaGqaaOGaa8xgGiaabohaaaa@3F6B@ (although the values of the r mse MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca qGYbGaeyOeI0IaaeyBaiaabohacaqGLbaaleqaaaaa@3D20@ of the estimators of τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa a@3A2A@ were large, those of the median of their absolute relative errors were not). The deviation from the assumed normal distribution of the effects β j ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHYoGyda qhaaWcbaGaamOAaaqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaaa aaa@3D9B@ caused that all the proposed estimators presented problems of overestimation. Neither of the two types of proposed estimators UMLEs and CMLEs performed uniformly better than the other, but the UMLEs performed in a greater number of cases slightly better than the CMLEs.

Table 6.2
Relative biases, square roots of relative mean square errors and medians of relative errors and absolute relative errors of the
estimators of the population sizes
Table summary
This table displays the results of Relative biases. The information is grouped by Population
Sampling rates
Estimator (appearing as row headers), I, II, III, IV, XXXX
0.51, XXXX
0.4 and XXXX
0.5, calculated using XXXX and XXXX
units of measure (appearing as column headers).
Population I II III IV
Sampling rates f 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIXaaabeaaaaa@39D4@
0.51
f 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIYaaabeaaaaa@39D5@
0.40
f 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIXaaabeaaaaa@39D4@
0.50
f 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIYaaabeaaaaa@39D5@
0.40
f 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIXaaabeaaaaa@39D4@
0.51
f 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIYaaabeaaaaa@39D5@
0.40
f 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIXaaabeaaaaa@39D4@
0.51
f 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIYaaabeaaaaa@39D5@
0.40
Estimator r b i a s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaabk haaeaacaqGIbaabaGaaeyAaaqaaiaabggaaeaacaqGZbaaaaa@3F1A@ r m s e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaafa qabeabbaaaaeaacaqGYbaabaGaaeyBaaqaaiaabohaaeaacaqGLbaa aaWcbeaaaaa@3BF1@ m d r e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2 gaaeaacaqGKbaabaGaaeOCaaqaaiaabwgaaaaa@3E2D@
m d a r e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2 gaaeaacaqGKbaabaGaaeyyaaqaaiaabkhaaeaacaqGLbaaaaa@3F12@ r b i a s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaabk haaeaacaqGIbaabaGaaeyAaaqaaiaabggaaeaacaqGZbaaaaa@3F1A@
r m s e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaafa qabeabbaaaaeaacaqGYbaabaGaaeyBaaqaaiaabohaaeaacaqGLbaa aaWcbeaaaaa@3BF1@ m d r e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2 gaaeaacaqGKbaabaGaaeOCaaqaaiaabwgaaaaa@3E2D@ m d a r e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2 gaaeaacaqGKbaabaGaaeyyaaqaaiaabkhaaeaacaqGLbaaaaa@3F12@ r b i a s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaabk haaeaacaqGIbaabaGaaeyAaaqaaiaabggaaeaacaqGZbaaaaa@3F1A@ r m s e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaafa qabeabbaaaaeaacaqGYbaabaGaaeyBaaqaaiaabohaaeaacaqGLbaa aaWcbeaaaaa@3BF1@ m d r e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2 gaaeaacaqGKbaabaGaaeOCaaqaaiaabwgaaaaa@3E2D@ m d a r e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2 gaaeaacaqGKbaabaGaaeyyaaqaaiaabkhaaeaacaqGLbaaaaa@3F12@ r b i a s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaabk haaeaacaqGIbaabaGaaeyAaaqaaiaabggaaeaacaqGZbaaaaa@3F1A@ r m s e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaafa qabeabbaaaaeaacaqGYbaabaGaaeyBaaqaaiaabohaaeaacaqGLbaa aaWcbeaaaaa@3BF1@ m d r e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2 gaaeaacaqGKbaabaGaaeOCaaqaaiaabwgaaaaa@3E2D@ m d a r e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdb9vqpue9WqpepGe9sr=x fr=xfr=xmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaab2 gaaeaacaqGKbaabaGaaeyyaaqaaiaabkhaaeaacaqGLbaaaaa@3F12@
Uncond. heter. MLEs  
τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D22@ -0.01 0.06 -0.01 0.04 -0.00 0.08 -0.00 0.05 0.10 0.11 0.10 0.10 -0 .04 6 0.20 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeylaiaabc dacaGGUaGaaGimaiaaisdadaqhaaWcbaGaaGOnaaqaaiaaicdacaGG UaGaaGOmaiaaicdaaaaaaa@3F5E@ 0.08 -0.03 0.05
τ ^ 2 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D23@ -0.06 0.26 -0.11 0.17 0.060.02 0.35 -0.01 0.16 0.16 0.43 0.07 0.18 0.04 2 15 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaGimaiaac6 cacaaIWaGaaGinamaaDaaaleaacaaIYaaabaGaaGymaiaaiwdaaaaa aa@3D49@ 2.2 -0.19 0.25
τ ^ ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaamaabmaabaGaamyvaaGaayjkaiaawMcaaaaaaaa@3C67@ -0.02 0.08 -0.03 0.05 0.010.02 0.10 0.01 0.06 0.11 0.15 0.10 0.10 -0 .02 7 15 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeylaiaabc dacaGGUaGaaGimaiaaikdadaqhaaWcbaGaaG4naaqaaiaaigdacaaI 1aaaaaaa@3DF5@ 0.55 -0.06 0.08
Cond. heter. MLEs  
τ ^ 1 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D10@ -0.00 0.07 -0.01 0.05 0.01 0.07 0.00 0.05 0.18 0.19 0.17 0.17 -0.051.6 0.09 -0.05 0.07
τ ^ 2 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D11@ -0.04 0.26 -0.09 0.17 0.09 0.38 0.01 0.16 0.18 0.46 0.10 0.18 0.12 2 21 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaGimaiaac6 cacaaIXaGaaGOmamaaDaaaleaacaaIYaaabaGaaGOmaiaaigdaaaaa aa@3D45@ 2.4 -0.14 0.23
τ ^ ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@ -0.01 0.08 -0.02 0.05 0.03 0.11 0.01 0.06 0.18 0.22 0.16 0.16 -0 .00 2 23 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeylaiaabc dacaGGUaGaaGimaiaaicdadaqhaaWcbaGaaGOmaaqaaiaaikdacaaI Zaaaaaaa@3DED@ 0.61 -0.06 0.08
Homogeneous MLEs  
τ ˜ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG aadaWgaaWcbaGaaGymaaqabaaaaa@3ABD@ -0.28 0.28 -0.28 0.28 -0.31 0.31 -0.31 0.31 -0.30 0.30 -0.30 0.30 -0.18 0.19 -0.18 0.18
τ ˜ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG aadaWgaaWcbaGaaGOmaaqabaaaaa@3ABE@ -0.40 0.40 -0.40 0.40 -0.40 0.40 -0.40 0.40 -0.40 0.40 -0.40 0.40 -0.30 0.32 -0.32 0.32
τ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG aaaaa@39D6@ -0.31 0.31 -0.31 0.31 -0.33 0.33 -0.33 0.33 -0.32 0.33 -0.32 0.32 -0.21 0.22 -0.21 0.21
Homogeneous BEs  
τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq badaWgaaWcbaGaaGymaaqabaaaaa@3AC9@ -0.28 0.28 -0.28 0.28 -0.31 0.31 -0.31 0.31 -0.30 0.30 -0.30 0.30 -0.18 0.19 -0.18 0.18
τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq badaWgaaWcbaGaaGOmaaqabaaaaa@3ACA@ -0.39 0.39 -0.39 0.39 -0.39 0.40 -0.39 0.39 -0.39 0.40 -0.39 0.39 -0.27 0.30 -0.29 0.29
τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq baaaa@39E2@ -0.31 0.31 -0.31 0.31 -0.33 0.33 -0.33 0.33 -0.32 0.32 -0.32 0.32 -0.20 0.21 -0.20 0.20

 

Figure 6.1 of section 6

Description for Figure 6.1

Figure showing six boxplot graphs for the values of the estimators of τ 1 , τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiaaigdaaeqaaOGaaiilaiabes8a0naaBaaaleaacaaIYaaa beaaaaa@3E0A@ and τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdqhaaa@39BC@ in Populations I and II. See Table 6.1 for the parameter values of the simulated populations. Each graph is made of four boxplots, one for each of UMLE, CMLE, MLE homogeneous and Bayes homogeneous. For both populations, the distributions of the estimators UMLE and CMLE were more or less symmetrical about τ 1 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiaaigdaaeqaaOGaaiOlaaaa@3B5F@ The distributions of the estimators MLE homogeneous and Bayes homogeneous are showing bias and a smaller dispersion. For both populations, the medians of the distributions of the estimators UMLE and CMLE are close to τ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiaaikdaaeqaaOGaaiOlaaaa@3B5F@ But, these distributions are skewed to the right with very long tails. The distributions of estimators MLE homogeneous and Bayes homogeneous are showing bias and a smaller dispersion. Similar conclusions can be drawn for τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdqhaaa@39BB@ boxplots.

 

Figure 6.2 of section 6

Description for Figure 6.2

Figure showing six boxplot graphs for the values of the estimators of τ 1 , τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiaaigdaaeqaaOGaaiilaiabes8a0naaBaaaleaacaaIYaaa beaaaaa@3E0A@ and τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdqhaaa@39BC@ in Populations III and IV. See Table 6.1 for the parameter values of the simulated populations. Each graph is made of four boxplots, one for each of UMLE, CMLE, MLE homogeneous and Bayes homogeneous. For Population III, the distributions of the estimators UMLE and CMLE show a bias for τ 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiaaigdaaeqaaOGaaiilaaaa@3B5D@ worse for CMLE. The distributions of the estimators MLE homogeneous and Bayes homogeneous are showing bias and a smaller dispersion. For both populations, for τ 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiaaikdaaeqaaOGaaiilaaaa@3B5D@ the distributions are skewed to the right with very long tails. This problem is worse for Population IV. The distributions of the estimators MLE homogeneous and Bayes homogeneous are showing bias and a smaller dispersion. Similar conclusions can be drawn for τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbba9=b0P0RWFb9fq0FXxbbf9Ff0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiXdqhaaa@39BB@ boxplots.

 

With regard to the estimators derived under the assumption of homogeneous p i j ( k ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaa0 baaSqaaiaadMgacaWGQbaabaWaaeWaaeaacaWGRbaacaGLOaGaayzk aaaaaOGaaiilaaaa@3E97@ both the MLEs τ ˜ 1 , τ ˜ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga acamaaBaaaleaacaaIXaaabeaakiaacYcacuaHepaDgaacamaaBaaa leaacaaIYaaabeaaaaa@3E96@ and τ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga acaaaa@3A39@ and the Bayesian assisted estimators τ 1 , τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga afamaaBaaaleaacaaIXaaabeaakiaacYcacuaHepaDgaafamaaBaaa leaacaaIYaaabeaaaaa@3EAE@ and τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga afaaaa@3A45@ showed very similar behavior which was characterized by serious problems of bias that deteriorated their performance.

Notice that the percentages of samples in which the proposed estimators were not computed because of numerical convergence problems, as well as the number of samples in which the values of the estimators exceeded 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaaIXaGaaG imamaaCaaaleqabaGaaGynaaaaaaa@3AC6@ and that not were used in calculating the reported results of r bias MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x fr=xfbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk HiTiaabkgacaqGPbGaaeyyaiaabohaaaa@3B6C@ and r mse MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9pC0xbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca qGYbGaeyOeI0IaaeyBaiaabohacaqGLbaaleqaaaaa@3D20@ because they would have been seriously affected, were not large, except in Population IV. As was indicated by a reviewer, computing the r bias MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x fr=xfbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk HiTiaabkgacaqGPbGaaeyyaiaabohaaaa@3B6C@ and r mse MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x fr=xfbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk HiTiaab2gacaqGZbGaaeyzaaaa@3A8F@ of an estimator using only its available values lower than 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaaIXaGaaG imamaaCaaaleqabaGaaGynaaaaaaa@3AC6@ favors the proposed estimators. We agree with that observation and for this reason we also reported measures of the performance of the estimators based on the medians of the relative errors and absolute relative errors which are robust to large values of the estimators. Thus, if we supposed that any time that an estimator was not computed its value had been very large and we computed the values of the measures of the performance of the estimators that are based on the medians using the complete set of observations the results would not have been different from those reported in Table 6.2, and our conclusions based on these measures would not have changed.

The results of the simulation study on the 95% CIs are shown in Table 6.3. The main outcomes are the following. All the PLCIs and adjusted PLCIs for τ 1 : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccaGG6aaaaa@3BD9@ the ones based on the UMLE τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaaaa@3D85@ and those based on the CMLE τ ^ 1 ( C ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaOGaaiilaaaa@3E2D@ showed good values of the cp in Population I. The adjusted PLCIs presented also good values of the cp in Population II, but not the unadjusted PLCIs whose values of the cp were relatively low. In Population III the values of the cp of all the PLCIs and adjusted PLCIs for τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaaaaa@3B11@ were low, whereas in Population IV the values were only slightly low. A good characteristic of these CIs was that they showed pretty acceptable values of their mrl and mdrl in each of the situations that were considered. The PLCI for τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaaaaa@3B12@ based on τ ^ 2 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaaaa@3D86@ and the one based on τ ^ 2 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIYaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaaaa@3D74@ presented acceptable values of the cp in all the populations, except in Population IV, where the values were something low. However, in all the cases the mrl and mdrl of these CIs were so large that they were not useful for making reasonable inferences. Both PLCI for τ : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDca GG6aaaaa@3AE8@ the one based on the UMLE τ ^ ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaCaaaleqabaWaaeWaaeaacaWGvbaacaGLOaGaayzkaaaaaaaa @3CCA@ and that based on the CMLE τ ^ ( C ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaCaaaleqabaWaaeWaaeaacaWGdbaacaGLOaGaayzkaaaaaOGa aiilaaaa@3D72@ performed acceptably well in Populations I and II, although the means of their relative lengths were large in Population II because this measure is not robust to great values of the lengths of the intervals. In the other populations these CIs showed problems of low coverage and/or large relative lengths; thus their performance was not good. Both types of adjusted PLCIs performed well only in Population I, in the other populations they presented large values of their relative lengths. Neither of the two types of CIs: the ones based on the UMLEs and those based on CMLEs performed uniformly better than the other, but those based on the UMLEs performed in a greater number of cases slightly better than those based on the CMLEs.

With respect to the bootstrap CIs, we have that each of the two types of CIs for τ 1 : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccaGG6aaaaa@3BD9@ the one based on τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaaaa@3D85@ and that based on τ ^ 1 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGdbaacaGLOaGaayzk aaaaaaaa@3D73@ performed well in Populations I, II and IV, although in this last population the values of their cp were slightly low. In Population III the values of their cp were very low because of the biases of the point estimators of τ 1 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccaGGUaaaaa@3BCD@ The two types of bootstrap CIs for τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaaaaa@3B12@ performed badly in all the populations because the values of their relative lengths were large. Finally, the two types of CIs for τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa a@3A2A@ performed in general well. Notice that the values of their mrl tended to be large because this measure is not robust to great values of the lengths, whereas the values of their mdrl were acceptable. Neither of the two types of bootstrap CIs performed uniformly better than the other, but the CIs based on the UMLEs performed in most cases better than those based on the CMLEs.

Table 6.3
Coverage probabilities and means and medians of relative lengths of the 95% confidence intervals for the
population sizes
Table summary
This table displays the results of Coverage probabilities and means and medians of relative lengths of the 95% confidence intervals for the
population sizes. The information is grouped by Population
Sampling rates
Conf. Interval (appearing as row headers), I, II, III, IV, XXXX
0.51, XXXX, XXXX
0.40 and XXXX
0.50, calculated using cp, mrl and mdrl units of measure (appearing as column headers).
Population I II III IV
Sampling rates f 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIXaaabeaaaaa@39D4@
0.51
This is an empty cell f 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIYaaabeaaaaa@39D5@
0.40
f 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIXaaabeaaaaa@39D4@
0.50
This is an empty cell f 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIYaaabeaaaaa@39D5@
0.40
f 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIXaaabeaaaaa@39D4@
0.51
This is an empty cell f 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIYaaabeaaaaa@39D5@
0.40
f 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIXaaabeaaaaa@39D4@
0.51
This is an empty cell f 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaaIYaaabeaaaaa@39D5@
0.40
Conf. Interval cp mrl mdrl cp mrl mdrl cp mrl mdrl cp mrl mdrl
PLCI-  τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D22@ 0.95 0.22 0.22 0.85 0.22 0.22 0.61 0.24 0.24 0.891.6 0.23 0.24
Adj-PLCI-  τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D22@ 0.94 0.24 0.24 0.98 0.42 0.41 0.69 0.27 0.26 0.901.6 0.25 0.26
PLCI-  τ ^ 2 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D23@ 0.94 1.4 0.98 0.95 2.82 1.3 0.95 2.6 1.6 0.7719 7.16 1.4
PLCI-  τ ^ ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaamaabmaabaGaamyvaaGaayjkaiaawMcaaaaaaaa@3C67@ 0.95 0.66 0.59 0.97 0.91 0.65 1.0 1.0 0.79 0.8621 2.11 0.65
Adj-PLCI-  τ ^ ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaamaabmaabaGaamyvaaGaayjkaiaawMcaaaaaaaa@3C67@ 0.92 0.75 0.62 1.07.0 5.833 2.1 1.00.20 1.71 0.87 0.9022 2.74 0.78
Bootstr-CI-  τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D22@ 0.94 0.23 0.23 0.94 0.33 0.31 0.59 0.25 0.25 0.890.40 0.24 0.25
Bootstr-CI-  τ ^ 2 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D23@ 0.87 1.4 0.86 0.97 3.9 1.6 0.97 4.7 2.1 0.8313 4.63 0.90
Bootstr-CI-  τ ^ ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaamaabmaabaGaamyvaaGaayjkaiaawMcaaaaaaaa@3C67@ 0.93 0.38 0.28 0.99 0.97 0.49 0.98 1.1 0.52 0.8913 1.23 0.31
PLCI-  τ ^ 1 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D10@ 0.96 0.24 0.23 0.91 0.25 0.24 0.76 0.31 0.29 0.902.1 0.25 0.25
Adj-PLCI-  τ ^ 1 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D10@ 0.95 0.26 0.25 0.99 0.44 0.42 0.81 0.33 0.32 0.902.1 0.27 0.27
PLCI-  τ ^ 2 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D11@ 0.94 1.4 0.98 0.95 2.72 1.3 0.95 2.5 1.6 0.8525 7.54 1.7
PLCI-  τ ^ ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@ 0.95 0.64 0.54 0.941.6 1.2 0.62 0.862.0 3.11 0.74 0.9330 2.91 0.84
Adj-PLCI-  τ ^ ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@ 0.96 0.82 0.59 1.07.6 7.233 2.6 0.903.8 3.66 1.2 0.9432 2.95 0.90
Bootstr-CI-  τ ^ 1 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D10@ 0.96 0.29 0.29 0.98 0.31 0.31 0.48 0.40 0.39 0.891.6 0.31 0.30
Bootstr-CI-  τ ^ 2 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D11@ 0.90 1.5 1.0 0.98 4.7 1.7 0.97 5.5 2.4 0.9424 3.46 1.3
Bootstr-CI-  τ ^ ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@ 0.95 0.44 0.34 1.00 1.1 0.49 0.96 1.3 0.62 0.9425 0.886 0.43
Wald-CI-  τ ˜ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG aadaWgaaWcbaGaaGymaaqabaaaaa@3ABD@ 0.00 0.09 0.09 0.00 0.08 0.08 0.00 0.08 0.08 0.08 0.13 0.13
Wald-CI-  τ ˜ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG aadaWgaaWcbaGaaGOmaaqabaaaaa@3ABE@ 0.00 0.17 0.16 0.00 0.17 0.16 0.00 0.16 0.16 0.18 0.34 0.31
Wald-CI-  τ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG aaaaa@39D6@ 0.00 0.08 0.08 0.00 0.07 0.07 0.00 0.07 0.07 0.03 0.13 0.13
Wald-CI-  τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq badaWgaaWcbaGaaGymaaqabaaaaa@3AC9@ 0.00 0.09 0.09 0.00 0.09 0.09 0.00 0.08 0.08 0.11 0.15 0.15
Wald-CI-  τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq badaWgaaWcbaGaaGOmaaqabaaaaa@3ACA@ 0.00 0.17 0.17 0.00 0.17 0.17 0.00 0.17 0.17 0.25 0.36 0.33
Wald-CI-  τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq baaaa@39E2@ 0.00 0.08 0.08 0.00 0.08 0.08 0.00 0.08 0.07 0.04 0.15 0.14

 

With regard to the CIs based on the point estimators derived under the homogeneity assumption, all of them showed null values of the cp, except in Population IV where the values were different from zero, but still too low. The bad performance of these CIs in terms of the cp was a result of the large biases of the point estimators. Thus, despite of the very small values of the r ml MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x fr=xfbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk HiTiaab2gacaqGSbaaaa@39A0@ of these intervals the very low values of their cp did not allow making reasonable inferences.

Observe that in the first three populations the percentages of samples in which the proposed CIs were not computed because of numerical convergence problems as well as the number of samples in which the values of their relative lengths exceeded 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaaIXaGaaG imamaaCaaaleqabaGaaGynaaaaaaa@3AC6@ and that not were used in calculating the reported results of the mrl because they would have been seriously affected, were not large (less than 4%), except in the case of the adjusted PLCIs based on τ ^ ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaCaaaleqabaWaaeWaaeaacaWGvbaacaGLOaGaayzkaaaaaaaa @3CCA@ and on τ ^ ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaCaaaleqabaWaaeWaaeaacaWGdbaacaGLOaGaayzkaaaaaaaa @3CB8@ which in Population II were not computed in about 7% of the samples and the means of their relative lengths were computed without using 33 values greater than 10 5 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaaIXaGaaG imamaaCaaaleqabaGaaGynaaaakiaac6caaaa@3B82@ However, in Population IV some percentages were close to 20% and others close to 30%. The large values of these percentages were, in part, consequence of the relatively large values of the percentages of samples in which the corresponding point estimators were not computed. It is clear that computing the measures of performance of a CI using only the cases in which the interval was obtained or computing the rml using only the samples in which the relative lengths were lower than 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaaIXaGaaG imamaaCaaaleqabaGaaGynaaaaaaa@3AC6@ favors the proposed CIs. However, notice that practically in all cases in which those percentages were large, say larger than 5%, both the mrl and the mdrl of the intervals were large enough that those intervals were not useful for making inferences. Therefore, if the performance of these CIs was not good under this favorable assessment, it will not be good under a fairer evaluation. The exceptions to this pattern were the PLCI based on τ ^ 1 ( U ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga qcamaaDaaaleaacaaIXaaabaWaaeWaaeaacaWGvbaacaGLOaGaayzk aaaaaOGaaiilaaaa@3E3F@ and the two types of bootstrap CIs for τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa a@3A2A@ which showed acceptable performance and large percentage of samples in which were not computed. So, the results of these CIs should be taken with reserve.

6.2 Population constructed using data from the Colorado Springs study on HIV/AIDS transmission

In this simulation study we constructed a population using data from the Colorado Springs study on heterosexual transmission of HIV/AIDS. As was indicated in the introduction to this paper, this epidemiological research was focused on a population of people who lived in the Colorado Springs metropolitan area from 1982 1992 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x fr=xfbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeymaiaabM dacaqG4aGaaeOmaiabgkHiTiaabgdacaqG5aGaaeyoaiaabkdaaaa@3C8D@ and who were at high risk of acquiring and transmitting HIV. That population included drug users, sex workers and their personal contacts, defined as those persons with whom they had close social, sexual or drug-associated relations. In that study, 595 initial responders were selected in a non-random fashion through a sexually transmitted disease clinic, a drug clinic, self-referral and street outreach. The responders were asked for a complete enumeration of their personal contacts and a total of 7,379 contacts who were not in the set of the initial responders were named and included in the study. In our simulation study the set U 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS baaSqaaiaaigdaaeqaaaaa@3A26@ was defined as the set of the 595 initial responders and, as in Félix-Medina and Monjardin (2010), they were grouped into N=105 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaey ypa0JaaGymaiaaicdacaaI1aaaaa@3C72@ clusters of sizes m i s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohaaaa@3C34@ generated by sampling from a zero-truncated negative binomial distribution with parameter of size 2.5 and probability 2 / 3 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaWcgaqaai aaikdaaeaacaaIZaaaaiaac6caaaa@3AA6@ The sample mean and variance of the 105 values m i s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohaaaa@3C34@ were 5.67 and 15.03, respectively. It is worth noting that most of the people who were assigned to the same cluster came from the same original source of recruitment. A person was defined to be linked to a cluster if he or she was a personal contact of at least one element in that cluster. Since, approximately 95% of the 7,379 contacts of the initial responders were linked to only one cluster, and this could affect the performance of the proposed estimators, in our study we defined the set U 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaS baaSqaaiaaikdaaeqaaaaa@3A27@ as the subset of the 7,379 contacts formed by the 415 persons who were linked to at least two clusters plus the 379 sex workers who were linked to only one cluster. Thus, τ 1 =595, τ 2 =794 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccqGH9aqpcaaI1aGaaGyoaiaaiwdacaGG SaGaeqiXdq3aaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaaG4naiaaiM dacaaI0aaaaa@4511@ and τ=1,389. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDcq GH9aqpcaqGXaGaaeilaiaabodacaqG4aGaaeyoaiaac6caaaa@3F72@ It is worth noting that this population is the same as the one called “reduced population” by Félix-Medina and Monjardin (2010).

We set the sizes of the initial samples selected from the population to n=25. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbGaey ypa0JaaGOmaiaaiwdacaGGUaaaaa@3C8B@ This value of n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbaaaa@3958@ yielded the sampling rates: f 1 =0.46 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGimaiaac6cacaaI0aGaaGOn aaaa@3E31@ and f 2 =0.37. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS baaSqaaiaaikdaaeqaaOGaeyypa0JaaGimaiaac6cacaaIZaGaaG4n aiaac6caaaa@3EE4@ The simulation experiment was carried out as the previous one, except that each time that the value m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS baaSqaaiaadMgaaeqaaaaa@3A71@ was contained in an initial sample, all the people linked to cluster i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbaaaa@3953@ were included in the sample. We used the same number of replications r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbaaaa@395C@ and the same number B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGcbaaaa@392C@ of bootstrap samples as those used in the previous study. In addition, the values of the parameters of the initial distributions that were used to construct the Bayesian-assisted estimators τ k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga afamaaBaaaleaacaWGRbaabeaaaaa@3B61@ and the value of q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGXbaaaa@395B@ used to compute the Gaussian quadrature formulas (3.3) and (3.4) were the same as those used in the previous study.

The results of the simulation study are shown in Table 6.4. We can see that among the proposed estimators of the population sizes, only the estimators of τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaaaaa@3B11@ did not present problems of bias nor problems of instability. The estimators of τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaaaaa@3B12@ and τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa a@3A2A@ exhibited serious problems of bias, particularly the estimators of τ 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaGccaGGSaaaaa@3BCC@ which affected their performance. As a result of the performance of the point estimators, only the adjusted PLCIs and bootstrap CIs for τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaaaaa@3B11@ performed acceptably well, although the values of the cp of the bootstrap CIs were slightly low. The unadjusted PLCIs for τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaaaaa@3B11@ showed low values of the cp because of the deviation from the assumed Poisson distribution of the M i s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohacaGGUaaaaa@3CC6@ The PLCIs and bootstrap CIs for τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaaaaa@3B12@ and τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa a@3A2A@ presented very large values of the mrl and mdrl that these intervals were not useful. Observe that the percentages of samples in which the proposed point and interval estimators of τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaaaaa@3B11@ were not computed because of numerical convergence problems were small (less than 1.2%). Therefore, they were virtually not favored by the evaluation procedure. In the case of the proposed point and intervals estimators of τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaaaaa@3B12@ and τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa a@3A2A@ those percentages were large. However, if their performance was not good under this favorable assessment, it will not be good under a fairer evaluation.

With regard to the point estimators derived under the homogeneity assumption, we have that the MLEs τ ˜ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga acamaaBaaaleaacaaIXaaabeaaaaa@3B20@ and τ ˜ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga acamaaBaaaleaacaaIYaaabeaaaaa@3B21@ showed problems of bias which affected their performance; however, the estimator τ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga acaaaa@3A39@ did not show problems of bias and its performance was acceptable. The small bias exhibited by this estimator might be explained by the fact that the negative bias of τ ˜ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga acamaaBaaaleaacaaIXaaabeaaaaa@3B20@ was canceled out by the positive bias of τ ˜ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga acamaaBaaaleaacaaIYaaabeaakiaac6caaaa@3BDD@ The Bayesian-assisted estimators performed similarly to the previous ones, although in this case the estimator τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHepaDga afamaaBaaaleaacaaIYaaabeaaaaa@3B2D@ of τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaaaaa@3B12@ showed only mild problems of bias. The Wald CIs based on the MLEs and on the Bayesian-assisted estimators showed low values of the cp. However, since the values of the r ml MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x fr=xfbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeOCaiabgk HiTiaab2gacaqGSbaaaa@39A0@ of these intervals were acceptable, the intervals for τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaaaaa@3B12@ and τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDaa a@3A2A@ might provide some information about these parameters.

Table 6.4
Simulation results obtained for estimators and confidence intervals in a population constructed using data from the Colorado Springs study – Point estimators
Table summary
This table displays the results of Résultats Simulation results obtained for estimators and confidence intervals in a population constructed using data from the Colorado Springs study – Point estimators. The information is grouped by Estimators (appearing as row headers) and Point estimators(appearing as column headers).
Estimator r bias MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x fr=xfbpdbeqabeWaceGabiqabeqabmqabeabbaGcbaGaaeOCaiabgk HiTiaabkgacaqGPbGaaeyyaiaabohaaaa@3D98@ r mse MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0xh9v8Wq0db9Fn0dbba9pw0lfr=x fr=xfbpdbeqabeWaceGabiqabeqabmqabeabbaGcbaWaaOaaaeaaca qGYbGaeyOeI0IaaeyBaiaabohacaqGLbaaleqaaaaa@3CD6@ mdre mdare
Uncond. heter. MLEs  
τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D22@ -0.00 1 0.03 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeylaiaabc dacaqGSaGaaeimaiaabcdadaqhaaWcbaGaaGymaaqaaiaabcdacaqG SaGaaeimaiaabodaaaaaaa@3F2D@ 0.10 -0.01 0.07
τ ^ 2 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D23@ 1.7 16 3.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeymaiaabY cacaqG3aWaa0baaSqaaiaaigdacaaI2aaabaGaae4maiaabYcacaqG 1aaaaaaa@3DE4@ 4.5 0.79 0.79
τ ^ ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaaiaaiIcacaWGvbGaaGykaaaaaaa@3C43@ 0.95 17 3.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeimaiaabY cacaqG5aGaaeynamaaDaaaleaacaaIXaGaaG4naaqaaiaabodacaqG SaGaaeynaaaaaaa@3E9E@ 2.6 0.46 0.46
Cond. heter. MLEs  
τ ^ 1 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D10@ 0.01 0.57 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeimaiaabY cacaqGWaGaaeymamaaDaaaleaaaeaacaqGWaGaaeilaiaabwdacaqG 3aaaaaaa@3DCB@ 0.12 0.01 0.08
τ ^ 2 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D11@ 1.7 10 4.7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeymaiaabY cacaqG3aWaa0baaSqaaiaaigdacaaIWaaabaGaaeinaiaabYcacaqG 3aaaaaaa@3DE0@ 4.5 0.80 0.80
τ ^ ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@ 0.96 10 5.2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeimaiaabY cacaqG5aGaaeOnamaaDaaaleaacaaIXaGaaGimaaqaaiaabwdacaqG SaGaaeOmaaaaaaa@3E96@ 2.6 0.46 0.46
Homogeneous MLEs  
τ ˜ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG aadaWgaaWcbaGaaGymaaqabaaaaa@3ABD@ -0.22 0.23 -0.22 0.22
τ ˜ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG aadaWgaaWcbaGaaGOmaaqabaaaaa@3ABE@ 0.21 0.34 0.16 0.18
τ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG aaaaa@39D6@ 0.02 0.17 -0.00 0.10
Homogeneous BEs  
τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq badaWgaaWcbaGaaGymaaqabaaaaa@3AC9@ -0.22 0.23 -0.22 0.22
τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq badaWgaaWcbaGaaGOmaaqabaaaaa@3ACA@ 0.12 0.22 0.10 0.13
τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq baaaa@39E2@ -0.02 0.13 -0.04 0.09

 

Table 6.4(cont.)
Simulation results obtained for estimators and confidence intervals in a population constructed using data from the Colorado Springs study – Confidence intervals
Table summary
This table displays the results of Résultats Simulation results obtained for estimators and confidence intervals in a population constructed using data from the Colorado Springs study – Confidence intervals. The information is grouped by Conf. interval (appearing as row headers) and Confidence intervals(appearing as column headers).
Conf. interval cp mdre mdare
PLCI- τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D22@ 0.75 0.24 0.24
Adj-PLCI- τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D22@ 0.95 0.41 0.41
PLCI- τ ^ 2 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D23@ 0.398.3 1018 3.7
PLCI- τ ^ (U) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaaiaacIcacaWGvbGaaiykaaaaaaa@3C37@ 0.838.6 5.76 2.1
Adj-PLCI- τ ^ (U) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaaiaacIcacaWGvbGaaiykaaaaaaa@3C37@ 0.9921 1147 7.5
Bootstr-CI- τ ^ 1 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D22@ 0.91 0.37 0.37
Bootstr-CI- τ ^ 2 ( U ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaamyvaaGaayjkaiaawMca aaaaaaa@3D23@ 0.863.2 1129 3.6
Bootstr-CI- τ ^ (U) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaaiaacIcacaWGvbGaaiykaaaaaaa@3C37@ 0.883.2 6.328 2.0
PLCI- τ ^ 1 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D10@ 0.811.2 0.30 0.27
Adj-PLCI- τ ^ 1 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D10@ 0.971.2 0.45 0.44
PLCI- τ ^ 2 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D11@ 0.3910 9.617 3.6
PLCI- τ ^ ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@ 0.8920 6.29 2.6
Adj-PLCI- τ ^ ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@ 1.027 1432 9.3
Bootstr-CI- τ ^ 1 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGymaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D10@ 0.861.2 0.35 0.35
Bootstr-CI- τ ^ 2 ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaqhaaWcbaGaaGOmaaqaamaabmaabaGaam4qaaGaayjkaiaawMca aaaaaaa@3D11@ 0.908.0 9.735 3.9
Bootstr-CI- τ ^ ( C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaamaabmaabaGaam4qaaGaayjkaiaawMcaaaaaaaa@3C55@ 0.919.2 5.934 2.2
Wald-CI- τ ˜ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG aadaWgaaWcbaGaaGymaaqabaaaaa@3ABD@ 0.06 0.16 0.16
Wald-CI- τ ˜ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG aadaWgaaWcbaGaaGOmaaqabaaaaa@3ABE@ 0.71 0.60 0.53
Wald-CI- τ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaG aaaaa@39D6@ 0.73 0.35 0.31
Wald-CI- τ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq badaWgaaWcbaGaaGymaaqabaaaaa@3AC9@ 0.13 0.22 0.22
Wald-CI- τ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq badaWgaaWcbaGaaGOmaaqabaaaaa@3ACA@ 0.72 0.45 0.43
Wald-CI- τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpC0xe9LqFf0xe9 vqaqFeFr0xbba9Fa0P0RWFb9fq0FXxbbf9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaq baaaa@39E2@ 0.70 0.27 0.26
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